# The Distribution of the first Eigenvalue Spacing at the Hard Edge of Random Hermitian Matrices.

## Nicholas Witte

**Abstract: **
The distribution for the first eigenvalue spacing at the finite, lower
endpoint of the spectrum of the Laguerre unitary ensemble of finite rank
random matrices is found in terms of the fifth Painlevé system and the
solution of its associated linear isomonodromic system. In particular
it is characterised by the polynomial solutions to the isomonodromic
equations which are also orthogonal with respect to a deformation of the
Laguerre weight. In the scaling to the hard edge regime we find an
analogous situation where a certain Painlevé III system and its
associated linear isomonodromic system characterise the scaled distribution.
We undertake extensive analytical studies of this system and use this
knowledge to accurately compute the distribution and its moments.

{Joint work with P. Forrester.)